OK, as an attempted fix, consider this new function F(x): 1. let continued fraction expansion of x be [a0; a1,a2,a3,a4,...] 2. consider the the a[j] which obey (i) a[j]>=4 (ii) min(a[j-1],a[j+1])>=2 Replace these a by either a+1 (if a=even) or a-1 (if a=odd), also known as "a XOR 1." 3. result is F(x). This function's range has no "holes." So perhaps it is continuous function. (I would think it obviously is continuous, but apparently there is something wrong with my intuition?) It maps rationals to rationals and quadratic irrationals to quadratic irrationals. It would seem to have Kth derivatives at any rational argument x, for any K=1,2,3... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)